Mie v1 - Portal
20
________________________________ MATLAB Functions for Mie Scattering and Absorption Version 1 Christian Mätzler ________________________________ Research Report No. 2002-08 June 2002 Institut für Angewandte Physik Mikrowellenabteilung __________________________________________________________ Sidlerstrasse 5 Tel. : +41 31 631 89 11 3012 Bern Fax. : +41 31 631 37 65 Schweiz E-mail : [email protected]
Transcript of Mie v1 - Portal
________________________________
MATLAB Functions for Mie Scattering and Absorption
Version 1
Christian Mätzler
________________________________
Research Report No. 2002-08 June 2002
Institut für Angewandte Physik Mikrowellenabteilung
__________________________________________________________
Sidlerstrasse 5 Tel. : +41 31 631 89 11 3012 Bern Fax. : +41 31 631 37 65 Schweiz E-mail : [email protected]
1
MATLAB Functions for Mie Scattering and Absorption
Christian Mätzler, Institute of Applied Physics, University of Bern, June 20021
List of Contents
Abstract ............................................................................................................. 1 1 Introduction .................................................................................................... 2
2 Formulas for a homogeneous sphere................................................................ 2
2.1 Mie coefficients and Bessel functions..............................................................................2
2.2 Mie efficiencies and cross sections ..................................................................................3
2.3 The scattered far field.......................................................................................................4
2.4 The internal field ..............................................................................................................5
2.5 Computation of Qabs, based on the internal field..............................................................6
3 The MATLAB Programs .................................................................................... 7
3.1 Comments ........................................................................................................................7
3.2 The Function Mie_abcd ...................................................................................................8
3.3 The Function Mie .............................................................................................................8
3.4 The Function Mie_S12.....................................................................................................9
3.5 The Function Mie_xscan..................................................................................................10
3.6 The Function Mie_tetascan ..............................................................................................10
3.7 The Function Mie_pt........................................................................................................11
3.8 The Function Mie_Esquare ..............................................................................................11
3.9 The Function Mie_abs......................................................................................................12
4 Examples and Tests........................................................................................13
4.1 The situation of x=1, m=5+0.4i .......................................................................................13
4.2 Large size parameters.......................................................................................................15
4.3 Large refractive index ......................................................................................................16
5 Conclusion, and outlook to further developments ...........................................18 References.........................................................................................................18
Abstract
A set of Mie functions has been developed in MATLAB to compute the four Mie
coefficients an, bn, cn and dn, efficiencies of extinction, scattering, backscattering
and absorption, the asymmetry parameter, and the two angular scattering func-
tions S1 and S2. In addition to the scattered field, also the absolute-square of the internal field is computed and used to get the absorption efficiency in a way inde-
pendent from the scattered field. This allows to test the computational accuracy.
This first version of MATLAB Mie Functions is limited to homogeneous dielectric
spheres without change in the magnetic permeability between the inside and out-
side of the particle. Required input parameters are the complex refractive index, m= m’+ im”, of the sphere (relative to the ambient medium) and the size parameter,
x=ka, where a is the sphere radius and k the wave number in the ambient medium.
1 Corrections: Equation p. 16, April 2006, printing error in Equation (4.50) p. 5, March 2012
2
1 Introduction
This report is a description of Mie-Scattering and Mie-Absorption programs written
in the numeric computation and visualisation software, MATLAB (Math Works,
1992), for the improvement of radiative-transfer codes, especially to account for rain
and hail in the microwave range and for aerosols and clouds in the submillimeter, infrared and visible range. Excellent descriptions of Mie Scattering were given by
van de Hulst (1957) and by Bohren and Huffman (1983). The present programs are
related to the formalism of Bohren and Huffman (1983). In addition an extension
(Section 2.5) is given to describe the radial dependence of the internal electric field
of the scattering sphere and the absorption resulting from this field. Except for Section 2.5, equation numbers refer to those in Bohren and Huffman (1983), in
short BH, or in case of missing equation numbers, page numbers are given. For a
description of computational problems in the Mie calculations, see the notes on p.
126-129 and in Appendix A of BH.
2 Formulas for a homogeneous sphere
2.1 Mie coefficients and Bessel functions
MATLAB function: Mie_abcd
The key parameters for Mie calculations are the Mie coefficients an and bn to com-
pute the amplitudes of the scattered field, and cn and dn for the internal field,
respectively. The computation of these parameters has been the most challenging part in Mie computations due to the involvement of spherical Bessel functions up to
high order. With MATLAB’s built-in double-precision Bessel functions, the com-
putation of the Mie coefficients has so far worked well up to size parameters
exceeding 10’000; the coefficients are given in BH on p.100:
)]'()[()]'()[(
)]'()[()]'()[(
)]'()[()]'()[(
)]'()[()]'()[(
)1()1(
1
1
)1(
1
)1(2
1
2
mxmxjxhxxhmxj
mxmxjxjxxjmxjb
mxmxjxhxxhmxjm
mxmxjxjxxjmxjma
nnnn
nnnnn
nnnn
nnnnn
=
=
μ
μ
μ
μ
(4.53)
)]'()[()]'()[(
)]'()[()]'()[(
)]'()[()]'()[(
)]'()[()]'()[(
)1(
1
)1(2
)1(
1
)1(
1
)1()1(
1
)1(
1
)1(
1
mxmxjxhxxhmxjm
xxjxmhxxhxmjd
mxmxjxhxxhmxj
xxjxhxxhxjc
nnnn
nnnnn
nnnn
nnnnn
μ
μμ
μ
μμ
=
=
(4.52)
where m is the refractive index of the sphere relative to the ambient medium, x=ka
is the size parameter, a the radius of the sphere and k =2 / is the wave number
and the wavelength in the ambient medium. In deviation from BH, μ1 is the ratio
of the magnetic permeability of the sphere to the magnetic permeability of the
ambient medium (corresponding to μ1/μ in BH). The functions jn(z) and
)()1( zhn
=jn(z)+iyn(z) are spherical Bessel functions of order n (n= 1, 2,..) and of the
given arguments, z= x or mx, respectively, and primes mean derivatives with respect to the argument. The derivatives follow from the spherical Bessel functions them-
selves, namely
)()()]'([);()()]'([ )1()1(
1
)1(
1 znhzzhzzhznjzzjzzj nnnnnn == (p.127)
3
For completeness, the following relationships between Bessel and spherical Bessel
functions are given:
)(2
)( 5.0 zJz
zj nn += (4.9)
)(2
)( 5.0 zYz
zy nn += (4.10)
Here, J and Y are Bessel functions of the first and second kind. For n=0 and 1 the
spherical Bessel functions are given (BH, p. 87) by
zzzzzyzzzy
zzzzzjzzzj
/sin/cos)(;/cos)(
/cos/sin)(;/sin)(
2
10
2
10
==
==
and the recurrence formula
)(12
)()( 11 zfz
nzfzf nnn
+=+ + (4.11)
where fn is any of the functions jn and yn. Taylor-series expansions for small argu-ments of jn and yn are given on p. 130 of BH. The spherical Hankel functions are
linear combinations of jn and yn. Here, the first type is required
)()()()1( ziyzjzh nnn += (4.13)
The following related functions are also used in Mie theory (although we try to avoid
them here):
)()();()();()( )1( zzhzzzyzzzjz nnnnnn === (p.101, 183)
Often μ1=1; then, (4.52-4.53) simplify to
)]'()[()]'()[(
)]'()[()]'()[(
;)]'()[()]'()[(
)]'()[()]'()[(
)1()1(
)1()1(2
2
mxmxjxhxxhmxj
mxmxjxjxxjmxjb
mxmxjxhxxhmxjm
mxmxjxjxxjmxjma
nnnn
nnnnn
nnnn
nnnnn
=
=
)]'()[()]'()[(
)]'()[()]'()[(
;)]'()[()]'()[(
)]'()[()]'()[(
)1()1(2
)1()1(
)1()1(
)1()1(
mxmxjxhxxhmxjm
xxjxmhxxhxmjd
mxmxjxhxxhmxj
xxjxhxxhxjc
nnnn
nnnnn
nnnn
nnnnn
=
=
The parameters used in radiative transfer depend on an and bn, but not on cn and dn. The latter coefficients are needed when the electric field inside the sphere is of
interest, e.g. to test the field penetration in the sphere, to study the distribution of
heat sources or to compute absorption. The absorption efficiency Qabs, however, can
also be computed from the scattered radiation, Equations (3.25), (4.61-62) to be
shown below.
2.2 Mie efficiencies and cross sections
MATLAB functions: Mie, Mie_xscan
4
The efficiencies Qi for the interaction of radiation with a scattering sphere of radius
a are cross sections i (called Ci in BH) normalised to the particle cross section, a2,
where i stands for extinction (i=ext), absorption (i=abs), scattering (i=sca), back-
scattering (i=b), and radiation pressure (i=pr), thus
2a
Q ii =
Energy conservation requires that
absscaext QQQ += , or absscaext
+= (3.25)
The scattering efficiency Qsca follows from the integration of the scattered power over
all directions, and the extinction efficiency Qext follows from the Extinction Theorem
(Ishimaru, 1978, p. 14, van de Hulst, 1957, p. 31), also called Forward-Scattering
Theorem, leading to:
=
++=1
22
2))(12(
2
n
nnsca banx
Q (4.61)
=
++=1
2)Re()12(
2
n
nnext banx
Q (4.62)
and Qabs follows from (3.25). All infinite series can be truncated after nmax terms. For this number Bohren and Huffman (1983) proposed the value
243/1
max++= xxn (p.477)
and this value is used here as well.
Furthermore, the asymmetry parameter g= cos indicates the average cosine of
the scattering angle with respect to power; it is used in Two-Stream Models (Mea-
dor and Weaver, 1980), and it is related to the efficiency Qpr of radiation pressure:
cosscaextpr QQQ = (p.120)
+
+++
+
+=
=
+
=
+
1
**
1
1
*
12)Re(
)1(
12)Re(
1
)2(4cos
n
nnnn
n
nnsca bann
nbbaa
n
nn
xQ
Finally, the backscattering efficiency Qb, applicable to monostatic radar, is given by
2
12
)()1)(12(1
=
+=n
nn
n
b banx
Q (p.122)
2.3 The scattered far field
MATLAB functions: Mie_S12, Mie_pt, Mie_tetascan
If the detailed shape of the angular scattering pattern is required, e.g. to get the
phase matrix or phase function for radiative-transfer calculations (Chandrasekhar,
1960), the scattering functions S1 and S2 are required. These functions describe the
scattered field Es. The scattered far field in spherical coordinates (Es , Es ) for a unit-
amplitude incident field (where the time variation exp(-i t) has been omitted) is
given by
5
)(cossin
)(coscos
1
2
Sikr
eE
Sikr
eE
ikr
s
ikr
s
=
=
(p.111)
with the scattering amplitudes S1 and S2
=
=
++
+=
++
+=
1
2
1
1
)()1(
12)(cos
;)()1(
12)(cos
n
nnnn
n
nnnn
bann
nS
bann
nS
(4.74)
Es is the scattered far-field component in the scattering plane, defined by the inci-
dent and scattered directions, and Es is the orthogonal component. The angle is
the angle between the incident electric field and the scattering plane. The functions
n(cos ) and n(cos ) describe the angular scattering patterns of the spherical har-
monics used to describe S1 and S2 and follow from the recurrence relations
121 )1(cos;1
cos1
12+==
nnnnnnnn
n
n
n
n (4.47)
starting with (Deirmendjian, 1969, p. 15)
)2cos(3;cos;0;cos3;1;0 210210 ======
2.4 The internal field
MATLAB function: presently, no direct function, but see Mie_Esquare
The internal field E1 for an incident field with unit amplitude is given by
( )= +
+=
1
)1(
1
)1(
11)1(
)12
n
nennondc
nn
nNME (4.40)
where the vector-wave harmonic fields are given in spherical (r, , ) coordinates by
Mo1n(1)
=
0
cos n (cos ) jn (rmx /a)
sin n (cos ) jn (rmx /a)
Ne1n(1)
=
n(n +1)cos sin n (cos )jn (rmx /a)rmx /a
cos n (cos )[rmx /ajn (rmx /a)]'
rmx /a
sin n (cos )[rmx /ajn (rmx)]'
rmx /a
(4.50)
and the coordinate system is defined as for the scattered field. The vector-wave
functions N and M are orthogonal with respect to integration over directions. Fur-
thermore for different values of n, the N functions are orthogonal, too, and the same
is true for the M functions.
6
2.5 Computation of Qabs, based on the internal field
MATLAB functions: Mie_Esquare, Mie_abs
The absorption cross section of a particle with dielectric (i.e. Ohmic) losses is given
by (Ishimaru, 1978, p. 17)
dVk
V
abs=
2
1" E (1)
where ” is the imaginary part of the relative dielectric constant of the particle (here
with respect to the ambient medium). Thanks to the orthogonality of the spherical vector-wave functions this integral becomes in spherical coordinates
( )=
+
++++=1
1
1 0
222 )()()(cos"n
a
rnnabsnnndmmcdrrdk (2)
and the integration over azimuth has already been performed, leading to the fac-
tor . The functions in the integrand are absolute-square values of the series terms
of the components of the vector-waves (4.50)
( )
( ) 22
2
2
2
22
22
22
')()(cos
')()(cos
)()(cossin
)()(cos
)()(cos
z
zzjgn
z
zzjgn
z
zjgn
zjgm
zjgm
nnn
nnn
nnnr
nnn
nnn
=
=
=
=
=
(3)
Here z=mrk, and gn stands for
2
)1(
12
+
+=
nn
ngn
(4)
For the integrals over cos , analytic solutions can be obtained. First, from BH we
find
( )12
)1(2)(cos)(cos)(cos
221
1
22
+
+=+
n
nnd
nn (p.103)
and second, from (4.46) in BH and Equation 8.14.13 of Abramowitz and Stegun
(1965), we get
( ) ( )12
)1(2)(cos)(cos)(cos)(cossin
1
1
21
1
1
22
+
+==
n
ndPd
nn (5)
leading to the two parts (6) and (7) of the angular integral in (2)
( ) 21
1
)()12(2)(cos zjndmmm nn +=+= (6)
7
( )+++=++=
221
1
')()()1()12(2)(cos)(
z
zzj
z
zjnnndnnnn nn
rn (7)
Now, the absorption cross section follows from integration over the radial distance r
inside the sphere up to the sphere radius a:
( )=
+=1 0
222
"
n
a
nnnnabsdrrdncmk (8)
The integrand contains the radial dependence of the absolute-square electric field 2
E averaged over spherical shells (all and , constant r):
( )=
+=1
222
4
1
n
nnnndncmE (9)
and in terms of this quantity, the absorption efficiency becomes
=
x
abs dxxx
Q0
22
2''
"4E (10)
where x’=rk=z/m. Note that (9) is dimensionless because of the unit-amplitude
incident field; In case of Rayleigh scattering (x<<1) the internal field is constant, and
the corresponding squared-field ratio (9) is given by
222
9
+m
(11)
This quantity can be used to test the accuracy of the function, Mie_Esquare, for
small size parameters. In addition, Equation (10) can be used to test the accuracy
of the computation of Qabs from the difference, Qext –Qsca (4.61-62). Finally, it should
be remembered that all infinite series can be terminated after nmax terms.
3 The MATLAB Programs
3.1 Comments
3.1.1 Restrictions
1. The present restrictions are situations with μ1=1; this means that the permittiv-
ity of the sphere relative to the ambient medium is given by =m2.
2. There is a maximum x value for stable and correct computation of Mie scatter-ing. For a small or moderate imaginary refractive index m”, the maximum size
parameter ranges somewhere between 104 and 105, however, for m”>>1, the
maximum size parameter is strongly diminished.
3.1.2 Computation of Bessel Functions The ordinary Bessel Functions J (z) and Y (z) are standard functions in MATLAB.
The spherical Bessel Functions used here follow from (4.9-10) of BH.
3.1.3 Computation of angular functions
The angular functions, n and n, are computed from the recurrence relations (4.47)
of BH with the initial functions given for n=1 and 2.
8
3.1.4 Optimisation strategy
The programs were optimised rather with respect to computation speed than
memory space; all required function values are computed only once and then stored
for further use in vectors of dimension nmax.
3.2 The Function Mie_abcd
The following text lists the basic program to compute the Mie Coefficients an, bn,
cn, dn and to produce a matrix of nmax column vectors [an; bn; cn; dn]:
function result = Mie_abcd(m, x)
% Computes a matrix of Mie coefficients, a_n, b_n, c_n, d_n,
% of orders n=1 to nmax, complex refractive index m=m'+im",
% and size parameter x=k0*a, where k0= wave number
% in the ambient medium, a=sphere radius; % p. 100, 477 in Bohren and Huffman (1983) BEWI:TDD122
% C. Mätzler, June 2002
nmax=round(2+x+4*x^(1/3));
n=(1:nmax); nu = (n+0.5); z=m.*x; m2=m.*m; sqx= sqrt(0.5*pi./x); sqz= sqrt(0.5*pi./z);
bx = besselj(nu, x).*sqx;
bz = besselj(nu, z).*sqz;
yx = bessely(nu, x).*sqx;
hx = bx+i*yx; b1x=[sin(x)/x, bx(1:nmax-1)];
b1z=[sin(z)/z, bz(1:nmax-1)];
y1x=[-cos(x)/x, yx(1:nmax-1)];
h1x= b1x+i*y1x;
ax = x.*b1x-n.*bx; az = z.*b1z-n.*bz;
ahx= x.*h1x-n.*hx;
an = (m2.*bz.*ax-bx.*az)./(m2.*bz.*ahx-hx.*az);
bn = (bz.*ax-bx.*az)./(bz.*ahx-hx.*az);
cn = (bx.*ahx-hx.*ax)./(bz.*ahx-hx.*az); dn = m.*(bx.*ahx-hx.*ax)./(m2.*bz.*ahx-hx.*az);
result=[an; bn; cn; dn];
3.3 The Function Mie
The following text lists the Program to compute the Mie Efficiencies:
function result = Mie(m, x)
% Computation of Mie Efficiencies for given % complex refractive-index ratio m=m'+im"
% and size parameter x=k0*a, where k0= wave number in ambient
% medium, a=sphere radius, using complex Mie Coefficients
% an and bn for n=1 to nmax,
% s. Bohren and Huffman (1983) BEWI:TDD122, p. 103,119-122,477. % Result: m', m", x, efficiencies for extinction (qext),
% scattering (qsca), absorption (qabs), backscattering (qb),
% asymmetry parameter (asy=<costeta>) and (qratio=qb/qsca).
9
% Uses the function "Mie_abcd" for an and bn, for n=1 to nmax.
% C. Mätzler, May 2002.
if x==0 % To avoid a singularity at x=0
result=[real(m) imag(m) 0 0 0 0 0 0 1.5]; elseif x>0 % This is the normal situation
nmax=round(2+x+4*x^(1/3));
n1=nmax-1;
n=(1:nmax);cn=2*n+1; c1n=n.*(n+2)./(n+1); c2n=cn./n./(n+1);
x2=x*x; f=mie_abcd(m,x);
anp=(real(f(1,:))); anpp=(imag(f(1,:)));
bnp=(real(f(2,:))); bnpp=(imag(f(2,:)));
g1(1:4,nmax)=[0; 0; 0; 0]; % displaced numbers used for
g1(1,1:n1)=anp(2:nmax); % asymmetry parameter, p. 120 g1(2,1:n1)=anpp(2:nmax);
g1(3,1:n1)=bnp(2:nmax);
g1(4,1:n1)=bnpp(2:nmax);
dn=cn.*(anp+bnp);
q=sum(dn); qext=2*q/x2;
en=cn.*(anp.*anp+anpp.*anpp+bnp.*bnp+bnpp.*bnpp);
q=sum(en);
qsca=2*q/x2;
qabs=qext-qsca; fn=(f(1,:)-f(2,:)).*cn;
gn=(-1).^n;
f(3,:)=fn.*gn;
q=sum(f(3,:));
qb=q*q'/x2; asy1=c1n.*(anp.*g1(1,:)+anpp.*g1(2,:)+bnp.*g1(3,:)+bnpp.*g1(4,:));
asy2=c2n.*(anp.*bnp+anpp.*bnpp);
asy=4/x2*sum(asy1+asy2)/qsca;
qratio=qb/qsca;
result=[real(m) imag(m) x qext qsca qabs qb asy qratio]; end;
3.4 The Function Mie_S12
The following text lists the program to compute the two complex scattering ampli-
tudes S1 and S2:
function result = Mie_S12(m, x, u)
% Computation of Mie Scattering functions S1 and S2
% for complex refractive index m=m'+im", % size parameter x=k0*a, and u=cos(scattering angle),
% where k0=vacuum wave number, a=sphere radius;
% s. p. 111-114, Bohren and Huffman (1983) BEWI:TDD122
% C. Mätzler, May 2002
nmax=round(2+x+4*x^(1/3));
abcd=Mie_abcd(m,x);
an=abcd(1,:);
10
bn=abcd(2,:);
pt=Mie_pt(u,nmax);
pin =pt(1,:);
tin=pt(2,:);
n=(1:nmax); n2=(2*n+1)./(n.*(n+1));
pin=n2.*pin;
tin=n2.*tin;
S1=(an*pin'+bn*tin');
S2=(an*tin'+bn*pin'); result=[S1;S2];
3.5 The Function Mie_xscan
The following text lists the program to compute the a matrix of Mie efficiencies and
to plot them as a function of x:
function result = Mie_xscan(m, nsteps, dx)
% Computation and plot of Mie Efficiencies for given
% complex refractive-index ratio m=m'+im" % and range of size parameters x=k0*a,
% starting at x=0 with nsteps increments of dx
% a=sphere radius, using complex Mie coefficients an and bn
% according to Bohren and Huffman (1983) BEWI:TDD122
% result: m', m", x, efficiencies for extinction (qext), % scattering (qsca), absorption (qabs), backscattering (qb),
% qratio=qb/qsca and asymmetry parameter (asy=<costeta>).
% C. Mätzler, May 2002.
nx=(1:nsteps)'; x=(nx-1)*dx;
for j = 1:nsteps
a(j,:)=Mie(m,x(j));
end;
output_parameters='Real(m), Imag(m), x, Qext, Qsca, Qabs, Qb, <costeta>, Qb/Qsca'
m1=real(m);m2=imag(m);
plot(a(:,3),a(:,4:9)) % plotting the results
legend('Qext','Qsca','Qabs','Qb','<costeta>','Qb/Qsca')
title(sprintf('Mie Efficiencies, m=%g+%gi',m1,m2)) xlabel('x')
result=a;
3.6 The Function Mie_tetascan
The following text lists the program to compute the a matrix of Mie scattering
intensities 2
1S and
2
2S as a function of u=cos , and to display the result as a
polar diagram of with 2
1S in the upper half circle (0< < ) and
2
2S in the lower
half circle ( < <2 ). Both functions are symmetric with respect to both half circles:
function result = Mie_tetascan(m, x, nsteps)
11
% Computation and plot of Mie Power Scattering function for given
% complex refractive-index ratio m=m'+im", size parameters x=k0*a,
% according to Bohren and Huffman (1983) BEWI:TDD122
% C. Mätzler, May 2002.
nsteps=nsteps;
m1=real(m); m2=imag(m);
nx=(1:nsteps); dteta=pi/(nsteps-1);
teta=(nx-1).*dteta; for j = 1:nsteps,
u=cos(teta(j));
a(:,j)=Mie_S12(m,x,u);
SL(j)= real(a(1,j)'*a(1,j));
SR(j)= real(a(2,j)'*a(2,j)); end;
y=[teta teta+pi;SL SR(nsteps:-1:1)]';
polar(y(:,1),y(:,2))
title(sprintf('Mie angular scattering: m=%g+%gi, x=%g',m1,m2,x));
xlabel('Scattering Angle') result=y;
3.7 The Function Mie_pt
The following text lists the program to compute a matrix of n and n functions for
n=1 to nmax:
function result=Mie_pt(u,nmax)
% pi_n and tau_n, -1 <= u= cos <= 1, n1 integer from 1 to nmax
% angular functions used in Mie Theory % Bohren and Huffman (1983), p. 94 - 95
p(1)=1;
t(1)=u;
p(2)=3*u; t(2)=3*cos(2*acos(u));
for n1=3:nmax,
p1=(2*n1-1)./(n1-1).*p(n1-1).*u;
p2=n1./(n1-1).*p(n1-2);
p(n1)=p1-p2; t1=n1*u.*p(n1);
t2=(n1+1).*p(n1-1);
t(n1)=t1-t2;
end;
result=[p;t];
3.8 The Function Mie_Esquare
The following text lists the program to compute and plot the ( , ) averaged
absolute-square E-field as a function of x’=rk (for r<0<a):
function result = Mie_Esquare(m, x, nj)
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% Computation of nj+1 equally spaced values within (0,x)
% of the mean-absolute-square internal
% electric field of a sphere of size parameter x,
% complex refractive index m=m'+im",
% where the averaging is done over teta and phi, % with unit-amplitude incident field;
% Ref. Bohren and Huffman (1983) BEWI:TDD122,
% and my own notes on this topic;
% k0=2*pi./wavelength;
% x=k0.*radius; % C. Mätzler, May 2002
nmax=round(2+x+4*x^(1/3));
n=(1:nmax); nu =(n+0.5);
m1=real(m); m2=imag(m); abcd=Mie_abcd(m,x);
cn=abcd(3,:);dn=abcd(4,:);
cn2=abs(cn).^2;
dn2=abs(dn).^2;
dx=x/nj; for j=1:nj,
xj=dx.*j;
z=m.*xj;
sqz= sqrt(0.5*pi./z);
bz = besselj(nu, z).*sqz; % This is jn(z) bz2=(abs(bz)).^2;
b1z=[sin(z)/z, bz(1:nmax-1)]; % Note that sin(z)/z=j0(z)
az = b1z-n.*bz./z;
az2=(abs(az)).^2;
z2=(abs(z)).^2; n1 =n.*(n+1);
n2 =2.*(2.*n+1);
mn=real(bz2.*n2);
nn1=az2;
nn2=bz2.*n1./z2; nn=n2.*real(nn1+nn2);
en(j)=0.25*(cn2*mn'+dn2*nn');
end;
xxj=[0:dx:xj]; een=[en(1) en];
plot(xxj,een); legend('Radial Dependence of (abs(E))^2')
title(sprintf('Squared Amplitude E Field in a Sphere, m=%g+%gi x=%g',m1,m2,x))
xlabel('r k')
result=een;
3.9 The Function Mie_abs
The following text lists the program to compute the absorption efficiency, based
on Equation (9):
function result = Mie_abs(m, x)
% Computation of the Absorption Efficiency Qabs
% of a sphere of size parameter x,
13
% complex refractive index m=m'+im",
% based on nj internal radial electric field values
% to be computed with Mie_Esquare(nj,m,x)
% Ref. Bohren and Huffman (1983) BEWI:TDD122,
% and my own notes on this topic; % k0=2*pi./wavelength;
% x=k0.*radius;
% C. Mätzler, May 2002
nj=5*round(2+x+4*x.^(1/3))+160; e2=imag(m.*m);
dx=x/nj;
x2=x.*x;
nj1=nj+1;
xj=(0:dx:x); en=Mie_Esquare(m,x,nj);
en1=0.5*en(nj1).*x2; % End-Term correction in integral
enx=en*(xj.*xj)'-en1; % Trapezoidal radial integration
inte=dx.*enx;
Qabs=4.*e2.*inte./x2; result=Qabs;
4 Examples and Tests
4.1 The situation of x=1, m=5+0.4i
The execution of the command line
>> m =5 + 0.4i; x = 1; mie_abcd(m,x)
leads to column vectors [an; bn; cn; dn] for n=1 to nmax=7:
ans =
Columns 1 through 4 (for n=1 to 4)
0.2306 - 0.2511i 0.0055 - 0.0354i 0.0000 - 0.0008i 0.0000 - 0.0000i 0.0627 + 0.1507i 0.0159 + 0.0262i 0.0004 - 0.0006i 0.0000 - 0.0000i
-0.6837 + 0.0128i -0.3569 + 0.2199i 0.0752 + 0.0339i 0.0076 + 0.0000i
-1.1264 + 0.0093i 0.2135 + 0.1212i 0.0180 + 0.0003i 0.0023 - 0.0004i
Columns 5 through 7 (for n=5 to 7)
0.0000 - 0.0000i 0.0000 - 0.0000i 0.0000 - 0.0000i
0.0000 - 0.0000i 0.0000 - 0.0000i 0.0000 - 0.0000i
0.0010 - 0.0002i 0.0002 - 0.0000i 0.0000 - 0.0000i
0.0003 - 0.0001i 0.0001 - 0.0000i 0.0000 - 0.0000i
>> mie(m,x)
ans =
Columns 1 through 7 (for m’, m”, x, Qext, Qsca, Qabs, Qb)
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5.0000 0.4000 1.0000 1.9794 0.8795 1.0999 1.1138
Columns 8 through 9 (for <cos >, Qb/Qsca)
-0.0595 1.2664
To test the absorption efficiency, call Mie_abs:
>> Mie_abs(m,x)
ans =
1.0999
to confirm the result of mie(m,x), s. above.
Mie Efficiencies are plotted versus x (0 x 2) by Mie_xscan(m,201,0.01) in Fig. 1. Two
absorption peaks can be observed near x=0.6 and 0.85 and a rapid variation of the
scattering behaviour between the forward and backward hemisphere.
To plot the angular dependence of the scattered power in the two polarisations, the
function Mie_tetascan(m,x,201), for x=1, is used to provide Figure 2. Furthermore the absolute-square internal field is plotted versus the radial distance
in Figure 3 by calling Mie_Esquare (automatically called by Mie_abs).
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-0.5
0
0.5
1
1.5
2
2.5
3
3.5
4Mie Efficiencies, m=5+0.4i
QextQscaQabsQb<costeta>Qb/Qsca
Figure 1: Mie Efficiencies
for m=5+0.4i versus x,
output of the function,
Mie_xscan. Note the sign of <cos > changes a few
times between x 0.6 and
1.1.
15
0.1
0.2
0.3
0.4
30
210
60
240
90
270
120
300
150
330
180 0
Mie angular scattering: m=5+0.4i, x=1
Figure 2: Angular Mie-scattering dia-
gram of 2
1S (upper Half circle) and of
2
2S (lower half circle). Here, scatter-
ing in the backward hemisphere is
slightly larger than in the forward hemisphere (<cos > <0), a situation
mostly restricted to the region near
x=1.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.2
0.4
0.6
0.8
1
1.2
1.4Squared Amplitude E Field in a Sphere, m=5+0.4i, x=1
Radial Dependence of (abs(E))2
Figure 3: Radial variation of the ratio
of the internal/external absolute-
square electric field with parameters
given in the figure title. Here, the field is concentrated in the sphere centre.
To test the behaviour at low x values, Figure 1 indicates that the ratio Qb/Qsca
approaches 1.5 near x=0. This is exactly the value obtained for Rayleigh scattering (Ishimaru, 1978). For x=0.1 Equation (9) gives absolute square internal field values
between 0.0130 in the sphere centre and 0.0137 at the sphere surface, and for
decreasing x the absolute square internal field of 0.0122 of the Rayleigh Expression
(11) is reached at x=0.01. This means that the validity of the Rayleigh Approxima-
tion is quite limited, especially for large values of m.
4.2 Large size parameters
For x>>1, a large number of spherical harmonics have to be computed. Here we
consider the example of m=2+0.01i. Figure 4 shows the Mie Efficiencies over the x range from 0 to 25. As a result of the Extinction Paradox (van de Hulst, 1957), Qext
approaches the value 2 for very large x. The computed values are slightly higher in
Figure 4. For increasing x, Qext decreases to 2.09 at x=100, to 2.01 at x=2800 and
to Qext=2.0014 for x=40’000. The program still works at x=70'000 (Qext =2.0012),
but for x>80'000 NaN values are returned.
16
In addition, for x>>1, the backscatter efficiency Qb should approach the geometri-
cal-optics expression
Qb =m 1
m +1
2
which is 0.1111 in the present case. With m=2+0.01i, this value is obtained from the Function Mie for x > 300.
0 5 10 15 20 250
2
4
6
8
10
12
14Mie Efficiencies, m=2+0.01i
QextQscaQabsQb<costeta>Qb/Qsca
Figure 4: Mie Efficiencies for m=2+0.01i over the x range from 0 to 25. The
parameter with the maximum values and with the largest fluctuations is Qb. The curves of Qext and Qsca follow each other closely near values of 2. The lowest curve
shows Qabs, and <cos > approaches 1 for high x values.
4.3 Large refractive index
Problems arise for high values of m. This situation occurs for metallic spheres, e.g.
if m=50+50i, the maximum size parameter is about 13, and this value decreases
with increasing m. A plot of efficiencies with m=50+50i is shown in Figure 5, and
the radial dependence of the absolute-square E field is shown in Figure 6 for x =0.5; the skin effect is very pronounced already at this small x value. Furthermore, due to
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the acting depolarisation field, the internal field is very small even at the sphere
surface.
0 2 4 6 8 10 12-0.5
0
0.5
1
1.5
2
2.5
3
3.5
4Mie Efficiencies, m=50+50i
QextQscaQabsQb<costeta>Qb/Qsca
Figure 5: Mie efficiencies for a metal-like material (m=50+50i).
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50
0.5
1
1.5
2
2.5
3
3.5x 10-4 Squared Amplitude E Field in a Sphere, m=50+50i, x=0.5
Radial Dependence of (abs(E))2
Figure 6: Radial variation of the
ratio of the internal/external ab-
solute-square electric field for
m=50+50i, x=0.5. The skin effect
is clearly indicated by the con-centration of the field near the
sphere surface.
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5 Conclusion, and outlook to further developments
The presented software represents the first major programming effort by the author
in MATLAB; therefore the programs may not yet be optimum. Nevertheless, no
severe problems have been encountered so far. Indeed, MATLAB appears to be a
very efficient and accurate computing language suitable for the present application. However, further tests are required to get a more complete assessment of the Mie
functions. Improvements are required to allow Mie computations for metal-like
spheres over a larger x range than is presently possible.
Concerning further developments, the extension to situations with μ1 1 is straight-
forward. Furthermore, it would be nice to make better use of the existing MATLAB visualisation tools. Finally, an extension to include scattering on coated spheres
(BH, Section 8.1) and on other particles should be done as well.
References Abramowitz M. and I.A. Stegun (eds.), “Handbook of Mathematical Functions”, Dover Publication, New
York, NY (1965). Bohren C.F. and D.R. Huffman, “Absorption and Scattering of Light by Small Particles”, John Wiley,
New York, NY (1983). Chandrasekhar S., "Radiative Transfer", Dover Publication (1960), BEWI TDD 211. Deirmendjian, D. “Electromagnetic Scattering on Spherical Polydispersions”, American Elsevier, New
York, NY (1969). Ishimaru A., “Wave propagation and scattering in random media”, Vol. 1, Academic Press, Orlando,
FL (1978). Math Works, “MATLAB User’s Guide”, Natick, MA (1992). Meador W.E. and W.R. Weaver, “Two-Stream Approximations to Radiative Transfer in Planetary
Atmospheres: A Unified Description of Existing Methods and a New Improvement”, J. Atm. Sciences, Vol. 37, pp. 630-643 (1980).
van de Hulst H.C. “Light Scattering by Small Particles”, (1957), reprinted by Dover Publication, New York, NY (1981).
